Generic transversality of rigidity-dyad subspaces

Establish whether the subspaces generated by the rigidity dyads and supported response blocks are transverse in general, thereby determining when the passive mechanical network attains the reciprocity capacity bound for arbitrary graphs.

Background

The paper proves that the Jacobian rank deficit relative to the reciprocity ceiling equals the dimension of an explicit intersection, Y∩KV⊥K\mathcal{Y}\cap K V^{\perp}K, where VV is generated by the graph’s rigidity dyads. Numerical certificates establish attainment for many particular graphs and configurations, but these results do not provide a theorem covering all graphs.

A general result would characterize when the upper bound min⁡(nb,mTmS−p(p−1)/2)\min(n_b,m_Tm_S-p(p-1)/2) is attained and would connect the learning capacity of reciprocal mechanical networks to generic rigidity theory.

References

Whether they are transverse is a question about generic rigidity, and we leave it open.

— Reciprocity can halve what a mechanical network can learn  (2609.04169 - Vu et al., 3 Sep 2026) in Section 2, subsection “Attainment is a transversality condition”; Section 2, subsection “Certified attainment at rational configurations”; Limitations

Learning at several frequencies at once stays open. It stacks blocks from different $G(\omega)$ and may carry its own constraint. We have not looked at that.

— Reciprocity can halve what a mechanical network can learn  (2609.04169 - Vu et al., 3 Sep 2026) in Section 3, subsection “Numerical verification,” paragraph “And it survives at finite frequency”; Limitations

Whether a physical realisation, with noise, hysteresis and a bounded stiffness range, also settles on the floor is untested, though Theorems~\ref{thm:floor} and~\ref{thm:floorpd} bound it regardless, since both are properties of the reachable set.

— Reciprocity can halve what a mechanical network can learn  (2609.04169 - Vu et al., 3 Sep 2026) in Limitations, second item; Conclusion

Whether every physical realisation of non-reciprocity buys the same subspace is not settled here.

— Reciprocity can halve what a mechanical network can learn  (2609.04169 - Vu et al., 3 Sep 2026) in Section 3, subsection “Numerical verification,” paragraph “Which subspace non-reciprocity buys”; Limitations